Question: $ F = \left[\begin{array}{rr}3 & 0 \\ 2 & -2 \\ 1 & 5\end{array}\right]$ $ w = \left[\begin{array}{r}1 \\ 3\end{array}\right]$ What is $ F w$ ?
Because $ F$ has dimensions $(3\times2)$ and $ w$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ F w = \left[\begin{array}{rr}{3} & {0} \\ {2} & {-2} \\ \color{gray}{1} & \color{gray}{5}\end{array}\right] \left[\begin{array}{r}{1} \\ {3}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{3}\cdot{1}+{0}\cdot{3} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{3}\cdot{1}+{0}\cdot{3} \\ {2}\cdot{1}+{-2}\cdot{3} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{3}\cdot{1}+{0}\cdot{3} \\ {2}\cdot{1}+{-2}\cdot{3} \\ \color{gray}{1}\cdot{1}+\color{gray}{5}\cdot{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}3 \\ -4 \\ 16\end{array}\right] $